Problem 84 - College Algebra

Question

Use a graphing utility to graph each circle whose equation is given. $$ (y+1)^{2}=36-(x-3)^{2} $$

Step-by-Step Solution

Answer

The circle's center is at the point (3, -1) and it has a radius of 6.

Step 1: Identify the center of the circle

The given equation is \((y+1)^{2}=36-(x-3)^{2}\). Comparing with the standard form, we can see that \(h = 3\) and \(k = -1\), so the center of the circle is at the point (3, -1).

Step 2: Determine the radius of the circle

The equation can be rewritten as \((x-3)^{2} + (y+1)^{2} = 36\). From this form, we can identify that \(r^{2} = 36\). Hence, \(r = \sqrt{36} = 6\). This means the circle has a radius of 6.

Step 3: Graph the circle

Using a graphing tool, you can plot the circle by starting at the center (3, -1). From there, draw a circle with a radius of 6. Note that this will reach to points that are 6 units left, right, above, and below the center.

Key Concepts

Equation of a Circle

The equation of a circle is critical in understanding what a circle looks like on a graph. It's the mathematical blueprint that details its size and position. The general equation for a circle in Cartesian coordinates is \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) is the center of the circle, and \( r \) is its radius. In this particular exercise, given the equation \( (y+1)^2 = 36 - (x-3)^2 \), we reconfigure it to match the general form mentioned earlier. This shows us visually where the circle sits on the plane and how large it is.

Center of a Circle

The center of a circle is like its anchor point. From the standard form of a circle's equation, \( (x-h)^2 + (y-k)^2 = r^2 \), \( h \) and \( k \) represent the coordinates of the center. Extracting these values from the equation given in the exercise, \( (y+1)^2 + (x-3)^2 = 36 \), we get the center at \( (3, -1) \). When you're graphing, this is where you'll place your pencil first to start drawing the circle.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its edge and is a key determinant of the circle's size. According to our standard form, the radius \( r \) is gleaned by taking the square root of the right side of the equation. From the exercise, rearranging the given equation \( (y+1)^2 + (x-3)^2 = 36 \) allows us to see that the radius squared, \( r^2 \), is 36, suggesting that the radius \( r \) is 6. This measurement is essential for sketching the circle's circumference accurately.

Graphing Utility Usage

Modern mathematics heavily relies on technology, such as graphing utilities, to visualize complex equations. These tools take an algebraic expression, like the equation of a circle, and translate it into a visual graph. By entering the equation of our circle, the graphing utility maps out the precise shape based on the center and radius we've determined. This way, we can see an exact representation of the circle without manually plotting points, which is especially helpful for complex equations or when needing a quick, accurate reference for further calculations.